<s>
In	O
linear	B-Language
algebra	I-Language
,	O
an	O
alternant	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
formed	O
by	O
applying	O
a	O
finite	O
list	O
of	O
functions	O
pointwise	O
to	O
a	O
fixed	O
column	O
of	O
inputs	O
.	O
</s>
<s>
An	O
alternant	O
determinant	O
is	O
the	O
determinant	O
of	O
a	O
square	O
alternant	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
(	O
Some	O
authors	O
use	O
the	O
transpose	O
of	O
the	O
above	O
matrix	B-Architecture
.	O
)	O
</s>
<s>
The	O
alternant	O
can	O
be	O
used	O
to	O
check	O
the	O
linear	O
independence	O
of	O
the	O
functions	O
in	O
function	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Then	O
the	O
alternant	O
is	O
the	O
matrix	B-Architecture
and	O
the	O
alternant	O
determinant	O
is	O
Therefore	O
M	O
is	O
invertible	O
and	O
the	O
vectors	O
form	O
a	O
basis	O
for	O
their	O
spanning	O
set	O
:	O
in	O
particular	O
,	O
and	O
are	O
linearly	O
independent	O
.	O
</s>
<s>
Linear	O
dependence	O
of	O
the	O
columns	O
of	O
an	O
alternant	O
does	O
not	O
imply	O
that	O
the	O
functions	O
are	O
linearly	O
dependent	O
in	O
function	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
is	O
in	O
the	O
nullspace	B-Algorithm
of	O
the	O
matrix	B-Architecture
:	O
that	O
is	O
,	O
.	O
</s>
<s>
If	O
and	O
the	O
functions	O
are	O
all	O
polynomials	O
,	O
then	O
divides	O
the	O
alternant	O
determinant	O
for	O
all	O
In	O
particular	O
,	O
if	O
V	O
is	O
a	O
Vandermonde	O
matrix	B-Architecture
,	O
then	O
divides	O
such	O
polynomial	O
alternant	O
determinants	O
.	O
</s>
<s>
Alternant	O
matrices	O
are	O
used	O
in	O
coding	B-Error_Name
theory	I-Error_Name
in	O
the	O
construction	O
of	O
alternant	B-Error_Name
codes	I-Error_Name
.	O
</s>
